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OCR MEI S1 2009 January Q5

8 marks Moderate -0.8

5 Each day Anna drives to work.

$R$ is the event that it is raining.
$L$ is the event that Anna arrives at work late.

You are given that $\mathrm { P } ( R ) = 0.36 , \mathrm { P } ( L ) = 0.25$ and $\mathrm { P } ( R \cap L ) = 0.2$.

Determine whether the events $R$ and $L$ are independent.
Draw a Venn diagram showing the events $R$ and $L$. Fill in the probability corresponding to each of the four regions of your diagram.
Find $\mathrm { P } ( L \mid R )$. State what this probability represents.

OCR MEI S1 2009 January Q6

17 marks Easy -1.2

6 The temperature of a supermarket fridge is regularly checked to ensure that it is working correctly. Over a period of three months the temperature (measured in degrees Celsius) is checked 600 times. These temperatures are displayed in the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{7b92607f-1bf9-45f0-997b-fe76c88b5fcd-4_1054_1649_539_248}

Use the diagram to estimate the median and interquartile range of the data.
Use your answers to part (i) to show that there are very few, if any, outliers in the sample.
Suppose that an outlier is identified in these data. Discuss whether it should be excluded from any further analysis.
Copy and complete the frequency table below for these data.
Temperature
$( t$ degrees Celsius $)$
$3.0 \leqslant t \leqslant 3.4$ $3.4 < t \leqslant 3.8$ $3.8 < t \leqslant 4.2$ $4.2 < t \leqslant 4.6$ $4.6 < t \leqslant 5.0$
Frequency 243 157
Use your table to calculate an estimate of the mean.
The standard deviation of the temperatures in degrees Celsius is 0.379 . The temperatures are converted from degrees Celsius into degrees Fahrenheit using the formula $F = 1.8 C + 32$. Hence estimate the mean and find the standard deviation of the temperatures in degrees Fahrenheit.

OCR MEI S1 2009 January Q7

19 marks Standard +0.3

7 An online shopping company takes orders through its website. On average $80 \%$ of orders from the website are delivered within 24 hours. The quality controller selects 10 orders at random to check when they are delivered.

Find the probability that
(A) exactly 8 of these orders are delivered within 24 hours,
(B) at least 8 of these orders are delivered within 24 hours. The company changes its delivery method. The quality controller suspects that the changes will mean that fewer than $80 \%$ of orders will be delivered within 24 hours. A random sample of 18 orders is checked and it is found that 12 of them arrive within 24 hours.
Write down suitable hypotheses and carry out a test at the $5 \%$ significance level to determine whether there is any evidence to support the quality controller's suspicion.
A statistician argues that it is possible that the new method could result in either better or worse delivery times. Therefore it would be better to carry out a 2 -tail test at the $5 \%$ significance level. State the alternative hypothesis for this test. Assuming that the sample size is still 18, find the critical region for this test, showing all of your calculations.

OCR MEI S1 2016 June Q1

7 marks Moderate -0.8

1 The stem and leaf diagram illustrates the weights in grams of 20 house sparrows.

25	0
26	0	5	8
27	7	9
28	1	4	5
29	0	0	2
30	7	7
31	6
32	0	4	7
33	3	3

Key: $\quad 27 \quad \mid \quad 7 \quad$ represents 27.7 grams

Find the median and interquartile range of the data.
Determine whether there are any outliers.

OCR MEI S1 2016 June Q2

7 marks Moderate -0.8

2 In a hockey league, each team plays every other team 3 times. The probabilities that Team A wins, draws and loses to Team B are given below.

$\mathrm { P } ($ Wins $) = 0.5$
$\mathrm { P } ($ Draws $) = 0.3$
$\mathrm { P } ($ Loses $) = 0.2$

The outcomes of the 3 matches are independent.

Find the probability that Team A does not lose in any of the 3 matches.
Find the probability that Team A either wins all 3 matches or draws all 3 matches or loses all 3 matches.
Find the probability that, in the 3 matches, exactly two of the outcomes, 'Wins', 'Draws' and 'Loses' occur for Team A.

OCR MEI S1 2016 June Q3

6 marks Easy -1.3

There are 5 runners in a race. How many different finishing orders are possible? [You should assume that there are no 'dead heats', where two runners are given the same position.] For the remainder of this question you should assume that all finishing orders are equally likely.
The runners are denoted by $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }$. Find the probability that they either finish in the order ABCDE or in the order EDCBA.
Find the probability that the first 3 runners to finish are $\mathrm { A } , \mathrm { B }$ and C , in that order.
Find the probability that the first 3 runners to finish are $\mathrm { A } , \mathrm { B }$ and C , in any order.

OCR MEI S1 2016 June Q4

8 marks Moderate -0.3

4 The probability distribution of the random variable $X$ is given by the formula $$\mathrm { P } ( X = r ) = \frac { k } { r ( r - 1 ) } \text { for } r = 2,3,4,5,6 .$$

Show that the value of $k$ is 1.2 . Using this value of $k$, show the probability distribution of $X$ in a table.
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR MEI S1 2016 June Q5

8 marks Easy -1.3

5 Measurements of sunshine and rainfall are made each day at a particular weather station. For a randomly chosen day,

$R$ is the event that at least 1 mm of rainfall is recorded,
$S$ is the event that at least 1 hour of sunshine is recorded.

You are given that $\mathrm { P } ( R ) = 0.28 , \mathrm { P } ( S ) = 0.87$ and $\mathrm { P } ( R \cup S ) = 0.94$.

Find $\mathrm { P } ( R \cap S )$.
Draw a Venn diagram showing the events $R$ and $S$, and fill in the probability corresponding to each of the four regions of your diagram.
Find $\mathrm { P } ( R \mid S )$ and state what this probability represents in this context.

OCR MEI S1 2016 June Q6

18 marks Moderate -0.8

6 An online store has a total of 930 different types of women's running shoe on sale. The prices in pounds of the types of women's running shoe are summarised in the table below.

Price $( \pounds x )$	$10 \leqslant x \leqslant 40$	$40 < x \leqslant 50$	$50 < x \leqslant 60$	$60 < x \leqslant 80$	$80 < x \leqslant 200$
Frequency	147	109	182	317	175

Calculate estimates of the mean and standard deviation of the shoe prices.
Calculate an estimate of the percentage of types of shoe that cost at least $\pounds 100$.
Draw a histogram to illustrate the data. The corresponding histogram below shows the prices in pounds of the 990 types of men's running shoe on sale at the same online store. \includegraphics[max width=\textwidth, alt={}, center]{aff0c5b2-011b-49a0-bf05-6d905f890eba-4_643_1192_340_440}
State the type of skewness shown by the histogram for men's running shoes.
Martin is investigating the percentage of types of shoe on sale at the store that cost more than $\pounds 100$. He believes that this percentage is greater for men's shoes than for women's shoes. Estimate the percentage for men's shoes and comment on whether you can be certain which percentage is higher.
You are given that the mean and standard deviation of the prices of men's running shoes are $\pounds 68.83$ and $\pounds 42.93$ respectively. Compare the central tendency and variation of the prices of men's and women's running shoes at the store.

OCR MEI S1 2016 June Q7

18 marks Moderate -0.3

7 To withdraw money from a cash machine, the user has to enter a 4-digit PIN (personal identification number). There are several thousand possible 4-digit PINs, but a survey found that $10 \%$ of cash machine users use the PIN '1234'.

16 cash machine users are selected at random.
(A) Find the probability that exactly 3 of them use 1234 as their PIN.
(B) Find the probability that at least 3 of them use 1234 as their PIN.
(C) Find the expected number of them who use 1234 as their PIN. An advertising campaign aims to reduce the number of people who use 1234 as their PIN. A hypothesis test is to be carried out to investigate whether the advertising campaign has been successful.
Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
A random sample of 20 cash machine users is selected.
(A) Explain why the test could not be carried out at the $10 \%$ significance level.
(B) The test is to be carried out at the $k \%$ significance level. State the lowest integer value of $k$ for which the test could result in the rejection of the null hypothesis.
A new random sample of 60 cash machine users is selected. It is found that 2 of them use 1234 as their PIN. You are given that, if $X \sim \mathrm {~B} ( 60,0.1 )$, then (to 4 decimal places) $$\mathrm { P } ( X = 2 ) = 0.0393 , \quad \mathrm { P } ( X < 2 ) = 0.0138 , \quad \mathrm { P } ( X \leqslant 2 ) = 0.0530 .$$ Using the same hypotheses as in part (ii), carry out the test at the $5 \%$ significance level. \section*{END OF QUESTION PAPER}

OCR MEI FP3 2011 June Q1

24 marks Challenging +1.2

1 The points $\mathrm { A } ( 2 , - 1,3 ) , \mathrm { B } ( - 2 , - 7,7 )$ and $\mathrm { C } ( 7,5,1 )$ are three vertices of a tetrahedron ABCD .
The plane ABD has equation $x + 4 y + 7 z = 19$.
The plane ACD has equation $2 x - y + 2 z = 11$.

Find the shortest distance from $B$ to the plane $A C D$.
Find an equation for the line AD .
Find the shortest distance from C to the line AD .
Find the shortest distance between the lines $A D$ and $B C$.
Given that the tetrahedron ABCD has volume 20, find the coordinates of the two possible positions for the vertex $D$.

OCR MEI FP3 2011 June Q2

24 marks Challenging +1.8

2 A surface $S$ has equation $z = 8 y ^ { 3 } - 6 x ^ { 2 } y - 15 x ^ { 2 } + 36 x$.

Sketch the section of $S$ given by $y = - 3$, and sketch the section of $S$ given by $x = - 6$. Your sketches should include the coordinates of any stationary points but need not include the coordinates of the points where the sections cross the axes.
From your sketches in part (i), deduce that $( - 6 , - 3 , - 324 )$ is a stationary point on $S$, and state the nature of this stationary point.
Find $\frac { \partial z } { \partial x }$ and $\frac { \partial z } { \partial y }$, and hence find the coordinates of the other three stationary points on $S$.
Show that there are exactly two values of $k$ for which the plane with equation $$120 x - z = k$$ is a tangent plane to $S$, and find these values of $k$.

OCR MEI FP3 2011 June Q3

24 marks Challenging +1.8

1. Given that $y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }$, show that $1 + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = \left( \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } x } + \frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 2 } x } \right) ^ { 2 }$. The arc of the curve $y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }$ for $0 \leqslant x \leqslant \ln a$ (where $a > 1$ ) is denoted by $C$.
2. Show that the length of $C$ is $\frac { a - 1 } { \sqrt { a } }$.
3. Find the area of the surface formed when $C$ is rotated through $2 \pi$ radians about the $x$-axis.
An ellipse has parametric equations $x = 2 \cos \theta , y = \sin \theta$ for $0 \leqslant \theta < 2 \pi$.
1. Show that the normal to the ellipse at the point with parameter $\theta$ has equation $$y = 2 x \tan \theta - 3 \sin \theta$$
2. Find parametric equations for the evolute of the ellipse, and show that the evolute has cartesian equation $$( 2 x ) ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } = 3 ^ { \frac { 2 } { 3 } }$$
3. Using the evolute found in part (ii), or otherwise, find the radius of curvature of the ellipse
  (A) at the point $( 2,0 )$,
  (B) at the point $( 0,1 )$.

OCR MEI FP3 2011 June Q4

24 marks Challenging +1.8

Show that the set $G = \{ 1,3,4,5,9 \}$, under the binary operation of multiplication modulo 11 , is a group. You may assume associativity.
Explain why any two groups of order 5 must be isomorphic to each other. The set $H = \left\{ 1 , \mathrm { e } ^ { \frac { 2 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 4 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 6 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 8 } { 5 } \pi \mathrm { j } } \right\}$ is a group under the binary operation of multiplication of complex numbers.
Specify an isomorphism between the groups $G$ and $H$. The set $K$ consists of the 25 ordered pairs $( x , y )$, where $x$ and $y$ are elements of $G$. The set $K$ is a group under the binary operation defined by $$\left( x _ { 1 } , y _ { 1 } \right) \left( x _ { 2 } , y _ { 2 } \right) = \left( x _ { 1 } x _ { 2 } , y _ { 1 } y _ { 2 } \right)$$ where the multiplications are carried out modulo 11 ; for example, $( 3,5 ) ( 4,4 ) = ( 1,9 )$.
Write down the identity element of $K$, and find the inverse of the element $( 9,3 )$.
Explain why $( x , y ) ^ { 5 } = ( 1,1 )$ for every element $( x , y )$ in $K$.
Deduce that all the elements of $K$, except for one, have order 5. State which is the exceptional element.
A subgroup of $K$ has order 5 and contains the element (9, 3). List the elements of this subgroup.
Determine how many subgroups of $K$ there are with order 5 .

OCR MEI FP3 2011 June Q5

24 marks Challenging +1.2

5 In this question, give probabilities correct to 4 decimal places.
Alpha and Delta are two companies which compete for the ownership of insurance bonds. Boyles and Cayleys are companies which trade in these bonds. When a new bond becomes available, it is first acquired by either Boyles or Cayleys. After a certain amount of trading it is eventually owned by either Alpha or Delta. Change of ownership always takes place overnight, so that on any particular day the bond is owned by one of the four companies. The trading process is modelled as a Markov chain with four states, as follows. On the first day, the bond is owned by Boyles or Cayleys, with probabilities $0.4,0.6$ respectively.
If the bond is owned by Boyles, then on the next day it could be owned by Alpha, Boyles or Cayleys, with probabilities $0.07,0.8,0.13$ respectively. If the bond is owned by Cayleys, then on the next day it could be owned by Boyles, Cayleys or Delta, with probabilities $0.15,0.75,0.1$ respectively. If the bond is owned by Alpha or Delta, then no further trading takes place, so on the next day it is owned by the same company.

Write down the transition matrix $\mathbf { P }$.
Explain what is meant by an absorbing state of a Markov chain. Identify any absorbing states in this situation.
Find the probability that the bond is owned by Boyles on the 10th day.
Find the probability that on the 14th day the bond is owned by the same company as on the 10th day.
Find the day on which the probability that the bond is owned by Alpha or Delta exceeds 0.9 for the first time.
Find the limit of $\mathbf { P } ^ { n }$ as $n$ tends to infinity.
Find the probability that the bond is eventually owned by Alpha. The probabilities that Boyles and Cayleys own the bond on the first day are changed (but all the transition probabilities remain the same as before). The bond is now equally likely to be owned by Alpha or Delta at the end of the trading process.
Find the new probabilities for the ownership of the bond on the first day.

For $n$ an integer: P: $n$ is an even number
Q: $n$ is a multiple of 4
For triangle ABC : P: $\quad \mathrm { B }$ is a right-angle
Q: $\quad \mathrm { AB } ^ { 2 } + \mathrm { BC } ^ { 2 } = \mathrm { AC } ^ { 2 }$

OCR MEI C1 Q4

4 marks Moderate -0.8

4 Find the coefficient of $x ^ { 3 }$ in the expansion of $( 2 + 3 x ) ^ { 5 }$.

OCR MEI C1 Q5

4 marks Easy -1.5

5 Find the value of the following.

$\left( \frac { 1 } { 3 } \right) ^ { - 2 }$
$16 ^ { \frac { 3 } { 4 } }$

OCR MEI C1 Q6

5 marks Moderate -0.8

6 The line $L$ is parallel to $y = - 2 x + 1$ and passes through the point $( 5,2 )$.
Find the coordinates of the points of intersection of $L$ with the axes.

OCR MEI C1 Q7

5 marks Moderate -0.8

7 Express $x ^ { 2 } - 6 x$ in the form $( x - a ) ^ { 2 } - b$.
Sketch the graph of $y = x ^ { 2 } - 6 x$, giving the coordinates of its minimum point and the intersections with the axes.

OCR MEI C1 Q8

5 marks Moderate -0.8

8 Find, in the form $y = m x + c$, the equation of the line passing through $\mathrm { A } ( 3,7 )$ and $\mathrm { B } ( 5 , - 1 )$.
Show that the midpoint of AB lies on the line $x + 2 y = 10$.

OCR MEI C1 Q11

13 marks Moderate -0.3

11 A cubic polynomial is given by $\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 10 x + 8$.

Show that $( x - 1 )$ is a factor of $\mathrm { f } ( x )$. Factorise $\mathrm { f } ( x )$ fully.
Sketch the graph of $y = f ( x )$.
The graph of $y = \mathrm { f } ( x )$ is translated by $\binom { - 3 } { 0 }$. Write down an equation for the resulting graph. You need not simplify your answer.
Find also the intercept on the $y$-axis of the resulting graph.

OCR MEI C3 Q1

3 marks Easy -1.2

1 Solve the equation $| 3 x + 2 | = 1$.

Price \(( \pounds x )\)	\(10 \leqslant x \leqslant 40\)	\(40 < x \leqslant 50\)	\(50 < x \leqslant 60\)	\(60 < x \leqslant 80\)	\(80 < x \leqslant 200\)
Frequency	147	109	182	317	175

Questions — OCR MEI (4333 questions)

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OCR MEI S1 2009 January Q5

OCR MEI S1 2009 January Q6

OCR MEI S1 2009 January Q7

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